Generalized ensemble theory with non-extensive statistics

Abstract

The non-extensive canonical ensemble theory is reconsidered with the method of Lagrange multipliers by maximizing Tsallis entropy, with the constraint that the normalized term of Tsallis' q−average of physical quantities, the sum p q j , is independent of the probability p i for Tsallis parameter q. The self-referential problem in the deduced probability and thermal quantities in non-extensive statistics is thus avoided, and thermodynamical relationships are obtained in a consistent and natural way. We also extend the study to the non-extensive grand canonical ensemble theory and obtain the q-deformed Bose-Einstein distribution as well as the q-deformed Fermi-Dirac distribution. The theory is further applied to the generalized Planck law to demonstrate the distinct behaviors of the various generalized q-distribution functions discussed in literature.

Ideal Bose-gas in nonadditive statistics

“…It appears in particular that the link between the entropy, energy, and temperature cannot be trivially replicated from the Boltzmann-Gibbs statistics. Strategies to generalize the Bose-Einstein and Fermi-Dirac distributions also vary [7,9,18,19].…”

Generalized Entropies Depending Only on the Probability and Their Quantum Statistics

“…Furthermore, as far as the nonextensive quantum statistical mechanics is concerned, the generalized Bose Einstein distribution for bosons and Fermi Dirac distribution for fermions have been already studied, and it has been shown that a possible distribution function in nonextensive quantum statistics can be written asn l = 1/(e α+βE l 2−q ± 1), see [10]. In Figures 2 and 3 we show for different values of q the Tsallis Bose-Einstein and Fermi-Dirac distributions respectively.…”

“…A direct generalization is the proposal n(x) = 1 e x 2−q ±1 [10]. We will assume a similar "corresponding" expression n =…”

The nonextensive statistical ensembles with dual thermodynamic interpretations

“…They include the original method [1], un-normalized method [21], normalized method [22], and the optimal Lagrange multiplier (OLM) method [23]. To avoid the self-referential problem, recently one generalized version of OLM method was proposed by an assumption that the Tsallis factor is independent of probability distribution of each microscopic configuration [24]. In the fundamental nonextensive thermodynamics [25][26][27][28], there have been some obstacles in establishment of complete nonextensive thermodynamic formalism.…”